Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,11,14,8)(4,15,13,6)(5,16,12,9)(18,24)(19,27)(20,29)(26,31)(33,34,38,42)(35,43,44,39)(36,40,45,47)(37,41,46,48);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,11)(15,16)(17,18,23,24)(19,22,27,32)(20,30,29,25)(21,26,28,31)(35,44)(36,45)(39,43)(40,47);(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,26,30,31)(33,35)(34,39)(36,41)(37,40)(38,44)(42,43)(45,48)(46,47);(1,4,7,13)(2,6,10,15)(3,12,14,5)(8,9,11,16)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,30)(31,32)(33,36,38,45)(34,40,42,47)(35,41,44,48)(37,43,46,39);(1,5,7,12)(2,9,10,16)(3,13,14,4)(6,11,15,8)(17,21,23,28)(18,26,24,31)(19,20,27,29)(22,30,32,25);(1,6)(2,4)(3,9)(5,11)(7,15)(8,12)(10,13)(14,16)(17,22)(18,27)(19,24)(20,31)(21,30)(23,32)(25,28)(26,29)(33,37)(34,41)(35,40)(36,39)(38,46)(42,48)(43,45)(44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343814);